Statistically Meaningful Geometry and Gauge Symmetry Breaking: A Geometric Foundation for Scientific Discovery and Intelligence Emergence
This paper introduces Statistically Meaningful Geometry (SMG), modeling over-parameterized learning systems as infinite-dimensional non-parametric Orlicz fiber bundles. It proves that under persistent out-of-distribution stimuli, continuous optimization fails, unmodeled variance accumulates as Active Acausal Tension, triggering a Gauge Symmetry Break (GSB) registered as a discrete step-jump in Structural G-Entropy. SMG provides a parameter-free, falsifiable dashboard to mathematically certify true intelligence and transform AI for Science into an engine of autonomous paradigm shifts.
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[Submitted on 3 Jul 2026]
Title:Statistically Meaningful Geometry and Gauge Symmetry Breaking: A Geometric Foundation for Scientific Discovery and Intelligence Emergence
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Abstract:The rapid scaling of over-parameterized machine learning architectures, particularly LLMs, raises a profound crisis: do these systems exhibit genuine intelligence, or are they merely sophisticated statistical pattern matchers? Classical flat Euclidean statistics cannot differentiate continuous interpolation from the autonomous discovery of novel causal laws. To resolve this, we introduce Statistically Meaningful Geometry (SMG), a framework modeling over-parameterized learning systems as infinite-dimensional non-parametric Orlicz fiber bundles. We prove that under persistent out-of-distribution (OOD) stimuli governed by unmodeled causal mechanisms, continuous optimization fails. Unmodeled variance is rejected by the visible horizontal base manifold, leaking into the unobservable vertical fiber space and generating an accumulation of Active Acausal Tension. Driven by the statistical manifold's non-linear curvature, this tension inevitably strikes a conjugate focal boundary ($T_{\text{crit}} = \pi^2 / K_{\text{max}}$), triggering localized volumetric collapse and a catastrophic matrix singularity ($[G_f]^{-1} \to \infty$). We demonstrate this geometric breakdown acts as the strict non-equilibrium trigger for a Gauge Symmetry Break (GSB). The system purges hidden tension from unobservable gauge redundancies, spontaneously crystallizing a new, mathematically independent horizontal coordinate axis. This non-parametric phase transition registers as a discrete $+1.0$ integer step-jump in observable Structural G-Entropy. By decoupling parameter charts and subjecting emergent axes to a Minimal Energy Path Criterion and a Causal Invariance Filter, we distinguish genuine discovery from malignant hallucinations. Ultimately, SMG provides a parameter-free, falsifiable dashboard to mathematically certify true intelligence, transforming AI for Science into an engine of autonomous paradigm shifts.
Subjects:
Machine Learning (cs.LG)
Cite as: arXiv:2607.05436 [cs.LG]
(or arXiv:2607.05436v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2607.05436
arXiv-issued DOI via DataCite
Submission history
From: Howell Tong [view email] [v1] Fri, 3 Jul 2026 13:34:56 UTC (66 KB)
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